From bad workstation you can ping the server, by name or IP?
I have seen this type of problem that I think I know where I am pinging but DNS or other network Name Service is messing things up.
To verify the ping connect to the server and get the IP address it is using. From the bad workstation ping that IP and add the -a switch to have the name resolved. You could also try flushing the DNS cache in the workstaion and include the arp recored too:
Then check your NIC setting to see if the subnet mask or default gateway are correct.
You have not mentioned what version of SQL you have on the client and server, are they compatable?